Interval estimation is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter, in contrast to point estimation, which is a single number.

## Formula

${\mu = \bar x \pm Z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt n}}$

Where −

- ${\bar x}$ = mean
- ${Z_{\frac{\alpha}{2}}}$ = the confidence coefficient
- ${\alpha}$ = confidence level
- ${\sigma}$ = standard deviation
- ${n}$ = sample size

### Example

**Problem Statement:**

Suppose a student measuring the boiling temperature of a certain liquid observes the readings (in degrees Celsius) 102.5, 101.7, 103.1, 100.9, 100.5, and 102.2 on 6 different samples of the liquid. He calculates the sample mean to be 101.82. If he knows that the standard deviation for this procedure is 1.2 degrees, what is the interval estimation for the population mean at a 95% confidence level?

**Solution:**

The student calculated the sample mean of the boiling temperatures to be 101.82, with standard deviation ${\sigma = 0.49}$. The critical value for a 95% confidence interval is 1.96, where ${\frac{1-0.95}{2} = 0.025}$. A 95% confidence interval for the unknown mean.

\ = (101.82 – 0.96, 101.82 + 0.96) \\[7pt]

\ = (100.86, 102.78) }$

As the level of confidence decreases, the size of the corresponding interval will decrease. Suppose the student was interested in a 90% confidence interval for the boiling temperature. In this case, ${\sigma = 0.90}$, and ${\frac{1-0.90}{2} = 0.05}$. The critical value for this level is equal to 1.645, so the 90% confidence interval is

\ = (101.82 – 0.81, 101.82 + 0.81) \\[7pt]

\ = (101.01, 102.63)}$

An increase in sample size will decrease the length of the confidence interval without reducing the level of confidence. This is because the standard deviation decreases as n increases.

## Margin of Error

The margin of error ${m}$ of interval estimation is defined to be the value added or subtracted from the sample mean which determines the length of the interval:

${Z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt n}}$

Suppose in the example above, the student wishes to have a margin of error equal to 0.5 with 95% confidence. Substituting the appropriate values into the expression for ${m}$ and solving for n gives the calculation.

\ = {\frac{2.35}{0.5}^2} \\[7pt]

\ = {(4.7)}^2

\ = 22.09 }$

To achieve 95% interval estimation for the mean boiling point with total length less than 1 degree, the student will have to take 23 measurements.

Table of Contents

1.statistics adjusted rsquared

2.statistics analysis of variance

4.statistics arithmetic median

8.statistics best point estimation

9.statistics beta distribution

10.statistics binomial distribution

11.statistics blackscholes model

13.statistics central limit theorem

14.statistics chebyshevs theorem

15.statistics chisquared distribution

16.statistics chi squared table

17.statistics circular permutation

18.statistics cluster sampling

19.statistics cohens kappa coefficient

21.statistics combination with replacement

23.statistics continuous uniform distribution

24.statistics cumulative frequency

25.statistics coefficient of variation

26.statistics correlation coefficient

27.statistics cumulative plots

28.statistics cumulative poisson distribution

30.statistics data collection questionaire designing

31.statistics data collection observation

32.statistics data collection case study method

34.statistics deciles statistics

36.statistics exponential distribution

40.statistics frequency distribution

41.statistics gamma distribution

43.statistics geometric probability distribution

46.statistics gumbel distribution

49.statistics harmonic resonance frequency

51.statistics hypergeometric distribution

52.statistics hypothesis testing

53.statistics interval estimation

54.statistics inverse gamma distribution

55.statistics kolmogorov smirnov test

57.statistics laplace distribution

58.statistics linear regression

59.statistics log gamma distribution

60.statistics logistic regression

63.statistics means difference

64.statistics multinomial distribution

65.statistics negative binomial distribution

66.statistics normal distribution

67.statistics odd and even permutation

68.statistics one proportion z test

69.statistics outlier function

71.statistics permutation with replacement

73.statistics poisson distribution

74.statistics pooled variance r

75.statistics power calculator

77.statistics probability additive theorem

78.statistics probability multiplicative theorem

79.statistics probability bayes theorem

80.statistics probability density function

81.statistics process capability cp amp process performance pp

83.statistics quadratic regression equation

84.statistics qualitative data vs quantitative data

85.statistics quartile deviation

86.statistics range rule of thumb

87.statistics rayleigh distribution

88.statistics regression intercept confidence interval

89.statistics relative standard deviation

90.statistics reliability coefficient

91.statistics required sample size

92.statistics residual analysis

93.statistics residual sum of squares

94.statistics root mean square

96.statistics sampling methods

98.statistics shannon wiener diversity index

99.statistics signal to noise ratio

100.statistics simple random sampling

102.statistics standard deviation

103.statistics standard error se

104.statistics standard normal table

105.statistics statistical significance

108.statistics stem and leaf plot

109.statistics stratified sampling

112.statistics tdistribution table

113.statistics ti 83 exponential regression

114.statistics transformations

116.statistics type i amp ii errors

119.statistics weak law of large numbers